A376796 Table T(n, k) read by upward antidiagonals. T(n,1) = A376180, T(n,2) = A376180(A376180(n)), T(n,3) = A376180(A376180(A376180(n))) and so on.

1, 4, 1, 5, 3, 1, 3, 6, 5, 1, 6, 5, 2, 6, 1, 2, 2, 6, 4, 2, 1, 13, 4, 4, 2, 3, 4, 1, 12, 16, 3, 3, 4, 5, 3, 1, 14, 10, 8, 5, 5, 3, 6, 5, 1, 11, 9, 11, 12, 6, 6, 5, 2, 6, 1, 15, 15, 14, 15, 10, 2, 2, 6, 4, 2, 1, 10, 17, 17, 9, 17, 11, 4, 4, 2, 3, 4, 1, 16, 11, 18, 18, 14, 18, 15, 3, 3, 4, 5, 3, 1, 9, 8, 15, 7, 7, 9, 7, 17, 5, 5, 3, 6, 5, 1, 17, 14, 12, 17, 13, 13, 14, 13

Offset

1,2

Comments

The sequence A376180 generates infinite cyclic group under composition. The identity element is A000027.

Each column can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of the row number n in each column is the n-th pentagonal number, A000326(n+1).

Each column is an intra-block permutation of the positive integers.

Links

Boris Putievskiy, Table of n, a(n) for n = 1..9870

Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.

Eric Weisstein's World of Mathematics, Polygonal Number.

Index entries for sequences that are permutations of the natural numbers.

Formula

(T(1,k),T(2,k), ... T(A002411(n),k)) is a permutation of the integers from 1 to A002411(n). (T(1,k),T(2,k), ... T(A002411(n),k)) = (T(1,1),T(2,1), ... T(A002411(n),1))^k.

Example

Table begins:
  k =      1   2   3   4   5
-----------------------------------
  n =  1:  1,  1,  1,  1,  1, ...
  n =  2:  4,  3,  5,  6,  2, ...
  n =  3:  5,  6,  2,  4,  3, ...
  n =  4:  3,  5,  6,  2,  4, ...
  n =  5:  6,  2,  4,  3,  5, ...
  n =  6:  2,  4,  3,  5,  6, ...
Column k = 1 contains the start of A376180.
Ord(T(1,1),T(2,1), ... T(6,1)) = 5, ord(T(1,1),T(2,1), ... T(18,1)) = 10, ord(T(1,1),T(2,1), ... T(40,1)) = 60, ord(T(1,1),T(2,1), ... T(75,1)) = 420, ord(T(1,1),T(2,1), ... T(126,1)) = 7140, where ord is the order of the permutation.

Mathematica

a[n_]:=Module[{L, R, P, Result}, L=Ceiling[Max[x/.NSolve[x^2 (x+1)-2 n==0, x, Reals]]]; R=n-((L-1)^2)*L/2; P=Which[OddQ[R]&&OddQ[L*(3*L-1)/2], (L*(3*L-1)/2-R+1+1)/2, OddQ[R]&&EvenQ[L*(3*L-1)/2], (R+L*(3*L-1)/2+1)/2, EvenQ[R]&&OddQ[L*(3*L-1)/2], Ceiling[(L*(3*L-1)/2+1)/2]+R/2, EvenQ[R]&&EvenQ[L*(3*L-1)/2], Ceiling[(L*(3*L-1)/2+1)/2]-R/2];
Result=P+(L-1)^2*L/2; Result] (*A376180*)
composeSequence[a_, n_, k_]:=Nest[a, n, k]
Nmax=5; T=Table[composeSequence[a, n, k], {n, 1, Nmax}, {k, 1, Nmax}]; T

Crossrefs

Cf. A000326, A002411, A376180, A378198, A378626.

Sequence in context: A115210 A199150 A329215 * A214892 A030352 A212643

Adjacent sequences: A376793 A376794 A376795 * A376797 A376798 A376799

Keyword

nonn,tabl

Author

Boris Putievskiy, Dec 22 2024

Status

approved